Subgroups — #36370338

Order 22144.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012571	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012570	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012537	C₂	1	2	yes	maximal, complemented
4	#26012536	C₂	1	2	yes	maximal, complemented
5	#26012535	C₂	1	2	yes	maximal, complemented
6	#26012534	C₂	1	2	yes	maximal, complemented
7	#26012506	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#19149744	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₇₃⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#10621924	C₂	1	1	yes	maximal, characteristic, complemented
11	#26012290	C₂	1	1	yes	maximal, characteristic
12	#19149767	—	2	1	no	—
13	#19149767	—	2	1	no	—
14	#19149767	C₂²	1	2	yes	complemented
15	#19149767	C₂²	1	2	yes	complemented
16	#19149767	C₂²	1	2	yes	complemented
17	#19149766	—	2	2	no	—
18	#19149766	C₂²	1	2	yes	complemented
19	C₂×#14583836	—	2	1	no	—
20	C₂×#14583836	C₂²	1	1	yes	characteristic
21	D₁₇₃⋊M₁₆	—	2	1	no	—
22	D₁₇₃⋊M₁₆	—	2	2	no	—
23	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
24	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10621924	C₂²	1	1	yes	characteristic
26	#19149750	C₂²	1	2	yes	complemented
27	#19149749	C₂²	1	2	yes	complemented
28	#19149744	C₂²	1	1	yes	characteristic, complemented
29	#19149744	C₂²	1	1	yes	characteristic, complemented
30	#19149744	C₂²	1	2	yes	complemented
31	#19149744	C₂²	1	2	yes	complemented
32	#19149743	C₂²	1	1	yes	characteristic, complemented
33	#19149743	C₂²	1	1	yes	characteristic, complemented
34	#19149733	C₂²	1	2	yes	complemented
35	C₈×#10621924	C₂²	1	2	yes	complemented
36	C₂×#14583827	C₂²	1	1	yes	characteristic
37	#41×D₁₇₃	C₂²	1	1	yes	characteristic
38	#19149711	C₄	1	2	yes	complemented
39	#19149710	C₄	1	1	yes	characteristic, complemented
40	#19149707	C₄	1	1	yes	characteristic, complemented
41	#19149706	C₄	1	2	yes	complemented
42	#19149704	C₄	1	1	yes	characteristic, complemented
43	#19149702	C₂²	1	2	yes	—
44	#19149700	C₂²	1	2	yes	—
45	M₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
46	#137×C₁₇₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10621924	—	2	1	no	—
48	#14583836	—	2	2	no	—
49	#14583836	—	2	2	no	—
50	#14583836	—	2	2	no	—
51	#14583836	—	2	1	no	—
52	#14583836	D₄	1	1	yes	characteristic
53	#14583836	D₄	1	1	yes	characteristic
54	#14583836	C₂³	1	1	yes	characteristic
55	#14583836	C₂³	1	1	yes	characteristic
56	#14583836	C₂³	1	2	yes	—
57	#14583835	—	2	2	no	—
58	#14583835	—	2	1	no	—
59	#14583835	D₄	1	2	yes	—
60	D₁₇₃⋊C₈	—	2	2	no	—
61	D₁₇₃⋊C₈	—	2	2	no	—
62	D₁₇₃⋊C₈	—	2	2	no	—
63	D₁₇₃⋊C₈	—	2	1	no	—
64	D₁₇₃⋊C₈	C₂³	1	2	yes	—
65	C₂×#13063443	—	2	2	no	—
66	C₂×#13063443	—	2	1	no	—
67	C₄×#10621924	—	2	1	no	—
68	C₄×#10621924	C₂³	1	1	yes	characteristic
69	C₄×#10621924	C₂³	1	1	yes	characteristic
70	#14583827	—	2	1	no	—
71	#14583827	—	2	1	no	—
72	#14583827	C₂×C₄	1	2	yes	—
73	#14583827	C₂³	1	2	yes	—
74	#14583827	C₂³	1	1	yes	characteristic
75	#14583826	—	2	1	no	—
76	#14583826	C₂×C₄	1	1	yes	characteristic
77	#14583825	—	2	1	no	—
78	#14583825	C₂³	1	1	yes	characteristic
79	Q₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic
80	C₂×D₆₉₂	—	2	1	no	—
81	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
82	D₄×D₁₇₃	—	2	1	no	—
83	D₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄×D₁₇₃	—	2	1	no	—
85	C₂×C₄×D₁₇₃	—	2	1	no	—
86	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
87	#14583816	C₂×C₄	1	2	yes	complemented
88	#14583815	C₂×C₄	1	2	yes	complemented
89	#14583813	C₂×C₄	1	2	yes	complemented
90	#14583813	C₂×C₄	1	2	yes	complemented
91	D₁₃₈₄	C₂×C₄	1	2	yes	complemented
92	#14583811	C₂×C₄	1	2	yes	complemented
93	#14583810	C₂×C₄	1	1	yes	characteristic
94	#14583809	C₂×C₄	1	2	yes	—
95	C₈×D₁₇₃	C₂³	1	2	yes	—
96	#41×C₁₇₃	C₂×C₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
98	Q₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
99	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic
101	C₂×#10621924	—	2	1	no	—
102	C₂×#10621924	—	2	1	no	—
103	C₂×#10621924	—	4	2	no	—
104	#13063443	—	2	2	no	—
105	#13063443	—	2	1	no	—
106	#13063443	—	2	1	no	—
107	#13063443	C₂×D₄	1	2	yes	—
108	#13063443	C₂×D₄	1	2	yes	—
109	C₂²×D₁₇₃	—	2	1	no	—
110	C₂²×D₁₇₃	—	2	1	no	—
111	C₂²×D₁₇₃	#41	1	1	yes	characteristic
112	Dic₃₄₆	—	2	1	no	—
113	Dic₃₄₆	C₂²×C₄	1	1	yes	characteristic
114	D₆₉₂	—	2	2	no	—
115	D₆₉₂	—	2	1	no	—
116	D₆₉₂	—	2	2	no	—
117	D₆₉₂	—	2	1	no	—
118	D₆₉₂	C₂²×C₄	1	2	yes	—
119	D₆₉₂	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₇₃	—	2	1	no	—
121	C₂²⋊D₁₇₃	—	2	1	no	—
122	C₂²⋊D₁₇₃	—	2	1	no	—
123	C₂×Dic₁₇₃	—	2	1	no	—
124	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₇₃	—	2	1	no	—
126	C₄×D₁₇₃	—	2	1	no	—
127	C₄×D₁₇₃	—	2	2	no	—
128	C₄×D₁₇₃	—	2	1	no	—
129	C₄×D₁₇₃	—	2	2	no	—
130	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
132	C₄×D₁₇₃	#41	1	1	yes	characteristic
133	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
134	#13063436	C₂²×C₄	1	2	yes	—
135	Q₈×C₁₇₃	—	2	1	no	—
136	Q₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
137	Q₈×C₁₇₃	C₂²×C₄	1	2	yes	—
138	D₄×C₁₇₃	—	2	1	no	—
139	D₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇₃	—	2	1	no	—
141	C₂×C₄×C₁₇₃	—	2	1	no	—
142	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
143	C₈×C₁₇₃	C₂²×C₄	1	2	yes	—
144	C₂²×C₁₇₃	—	2	1	no	—
145	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×D₁₇₃	—	2	1	no	—
147	C₂×D₁₇₃	—	2	2	no	—
148	C₂×D₁₇₃	—	4	1	no	—
149	C₂×D₁₇₃	—	8	1	no	—
150	C₂×D₁₇₃	—	8	1	no	—
151	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
152	C₂×D₁₇₃	C₂×#41	1	1	yes	characteristic
153	#10621924	—	2	2	no	—
154	#10621924	—	4	1	no	—
155	C₄×C₁₇₃	—	2	1	no	—
156	C₄×C₁₇₃	—	2	2	no	—
157	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
158	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
159	Dic₁₇₃	—	2	1	no	—
160	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
161	Dic₁₇₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₁₇₃	—	2	1	no	—
163	C₂×C₁₇₃	—	4	1	no	—
164	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
165	D₁₇₃	—	2	1	no	—
166	D₁₇₃	—	2	1	no	—
167	D₁₇₃	—	4	1	no	—
168	D₁₇₃	—	4	2	no	—
169	C₁₇₃	#2956	1	1	yes	characteristic, complemented, Sylow
170	#2956	—	173	1	no	maximal, Sylow
171	#575	—	173	1	no	—
172	#568	—	173	1	no	—
173	#568	—	173	1	no	—
174	#444	—	173	2	no	—
175	#444	—	173	2	no	—
176	#443	—	173	2	no	—
177	#443	—	173	2	no	—
178	#430	—	173	1	no	—
179	C₂×#104	—	173	1	no	—
180	C₂×#104	—	173	1	no	—
181	C₄×M₁₆	—	173	1	no	—
182	C₂×#41	—	173	1	no	—
183	C₂×#41	—	173	1	no	—
184	#137	—	173	1	no	—
185	#137	—	173	2	no	—
186	#137	#10621924	1	1	yes	characteristic, complemented
187	AGL(1,ℤ/8)	—	173	1	no	—
188	AGL(1,ℤ/8)	—	173	2	no	—
189	AGL(1,ℤ/8)	—	173	1	no	—
190	#135	—	173	2	no	—
191	#135	—	173	2	no	—
192	#131	—	346	2	no	—
193	#131	—	346	1	no	—
194	#131	—	346	1	no	—
195	#131	—	173	2	no	—
196	#131	—	173	2	no	—
197	#131	—	173	2	no	—
198	#131	—	173	2	no	—
199	C₂×M₁₆	—	346	2	no	—
200	C₂×M₁₆	—	346	1	no	—
201	C₂×M₁₆	—	346	1	no	—
202	C₂×M₁₆	—	173	1	no	—
203	C₂×M₁₆	—	173	1	no	—
204	C₂×M₁₆	—	173	1	no	—
205	C₂×M₁₆	—	173	1	no	—
206	C₂×C₄²	—	173	1	no	—
207	#108	—	173	2	no	—
208	#108	—	173	2	no	—
209	#104	—	173	1	no	—
210	#104	—	173	1	no	—
211	#104	—	173	1	no	—
212	#104	—	173	2	no	—
213	#104	—	173	1	no	—
214	#104	—	173	2	no	—
215	#97	—	173	2	no	—
216	C₄×C₈	—	173	2	no	—
217	#41	—	346	1	no	—
218	#41	—	346	1	no	—
219	#41	—	346	1	no	—
220	#41	—	346	1	no	—
221	#41	—	173	2	no	—
222	#41	—	173	1	no	—
223	#41	—	173	2	no	—
224	#41	—	173	1	no	—
225	#41	—	173	1	no	—
226	#41	C₂×#10621924	1	1	yes	characteristic
227	C₂×Q₈	—	173	1	no	—
228	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic
229	C₂×D₄	—	346	1	no	—
230	C₂×D₄	—	346	1	no	—
231	C₂×D₄	—	173	1	no	—
232	C₂×D₄	—	173	1	no	—
233	C₂²×C₄	—	346	1	no	—
234	C₂²×C₄	—	346	1	no	—
235	C₂²×C₄	—	346	1	no	—
236	C₂²×C₄	—	173	1	no	—
237	Q₁₆	—	173	2	no	—
238	Q₁₆	C₂×#10621924	1	2	yes	complemented
239	SD₁₆	—	173	2	no	—
240	SD₁₆	—	173	2	no	—
241	SD₁₆	—	173	2	no	—
242	SD₁₆	C₂×#10621924	1	2	yes	complemented
243	D₈	—	173	2	no	—
244	D₈	—	173	2	no	—
245	M₁₆	—	346	1	no	—
246	M₁₆	—	346	1	no	—
247	M₁₆	—	346	2	no	—
248	M₁₆	—	346	2	no	—
249	M₁₆	—	346	2	no	—
250	M₁₆	—	346	2	no	—
251	M₁₆	—	173	1	no	—
252	M₁₆	—	173	1	no	—
253	M₁₆	—	173	2	no	—
254	M₁₆	—	173	2	no	—
255	M₁₆	—	173	1	no	—
256	M₁₆	—	173	1	no	—
257	M₁₆	—	173	1	no	—
258	M₁₆	—	173	2	no	—
259	M₁₆	C₂×#10621924	1	1	yes	characteristic
260	C₂×C₈	—	346	1	no	—
261	C₂×C₈	—	346	1	no	—
262	C₂×C₈	—	346	2	no	—
263	C₂×C₈	—	346	2	no	—
264	C₂×C₈	—	346	2	no	—
265	C₂×C₈	—	346	2	no	—
266	C₂×C₈	—	173	2	no	—
267	C₂×C₈	—	173	2	no	—
268	C₄²	—	346	1	no	—
269	C₄²	—	173	1	no	—
270	C₄²	—	173	1	no	—
271	C₂³	—	346	1	no	—
272	C₂³	—	346	1	no	—
273	C₂³	—	173	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	346	1	no	—
276	Q₈	—	173	1	no	—
277	Q₈	C₂²×#10621924	1	1	yes	characteristic
278	Q₈	C₂²×#10621924	1	2	yes	—
279	D₄	—	2	1	no	—
280	D₄	—	346	1	no	—
281	D₄	—	346	2	no	—
282	D₄	—	346	2	no	—
283	D₄	—	346	1	no	—
284	D₄	—	346	1	no	—
285	D₄	—	346	1	no	—
286	D₄	—	346	1	no	—
287	D₄	—	173	2	no	—
288	D₄	—	173	1	no	—
289	D₄	C₂²×#10621924	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	346	1	no	—
293	C₂×C₄	—	346	1	no	—
294	C₂×C₄	—	346	2	no	—
295	C₂×C₄	—	346	2	no	—
296	C₂×C₄	—	346	1	no	—
297	C₂×C₄	—	346	1	no	—
298	C₂×C₄	—	346	1	no	—
299	C₂×C₄	—	346	1	no	—
300	C₂×C₄	—	173	1	no	—
301	C₂×C₄	—	173	1	no	—
302	C₂×C₄	—	173	1	no	—
303	C₂×C₄	—	173	1	no	—
304	C₂×C₄	—	173	1	no	—
305	C₂×C₄	—	692	2	no	—
306	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
307	C₈	—	346	2	no	—
308	C₈	—	346	1	no	—
309	C₈	—	346	1	no	—
310	C₈	—	173	2	no	—
311	C₈	—	173	2	no	—
312	C₈	—	173	2	no	—
313	C₈	C₂²×#10621924	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	346	2	no	—
316	C₂²	—	346	1	no	—
317	C₂²	—	1384	1	no	—
318	C₂²	—	1384	1	no	—
319	C₂²	—	173	1	no	—
320	C₂²	—	173	1	no	—
321	C₂²	—	692	1	no	—
322	C₂²	D₄×#10621924	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	2	no	—
325	C₄	—	346	2	no	—
326	C₄	—	346	1	no	—
327	C₄	—	173	1	no	—
328	C₄	—	173	1	no	—
329	C₄	—	692	1	no	—
330	C₄	D₄×#10621924	1	1	yes	characteristic
331	C₄	C₂³×#10621924	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	1	no	—
334	C₂	—	346	1	no	—
335	C₂	—	346	1	no	—
336	C₂	—	692	2	no	—
337	C₂	—	692	1	no	—
338	C₂	C₂×D₄×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10621924
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₄	C₂³×#10621924
Fitting	#137×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	#26012290	C₂
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	#137	#10621924
hypocenter	C₁₇₃	#2956
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #26012571	C₂	semidirect
2. #26012570	C₂	semidirect
3. #26012537	C₂	semidirect
4. #26012536	C₂	semidirect
5. #26012535	C₂	semidirect
6. #26012534	C₂	semidirect
7. #26012506	C₂	semidirect
8. C₂×#19149744	C₂	semidirect
9. D₁₇₃⋊#104	C₂	semidirect
10. M₁₆×#10621924	C₂	semidirect
14. #19149767	C₂²	semidirect
15. #19149767	C₂²	semidirect
16. #19149767	C₂²	semidirect
18. #19149766	C₂²	semidirect
26. #19149750	C₂²	semidirect
27. #19149749	C₂²	semidirect
28. #19149744	C₂²	semidirect
29. #19149744	C₂²	semidirect
30. #19149744	C₂²	semidirect
31. #19149744	C₂²	semidirect
32. #19149743	C₂²	semidirect
33. #19149743	C₂²	semidirect
34. #19149733	C₂²	semidirect
35. C₈×#10621924	C₂²	semidirect
38. #19149711	C₄	semidirect
39. #19149710	C₄	semidirect
40. #19149707	C₄	semidirect
41. #19149706	C₄	semidirect
42. #19149704	C₄	semidirect
46. #137×C₁₇₃	C₄	semidirect
87. #14583816	C₂×C₄	semidirect
88. #14583815	C₂×C₄	semidirect
89. #14583813	C₂×C₄	semidirect
90. #14583813	C₂×C₄	semidirect
91. D₁₃₈₄	C₂×C₄	semidirect
92. #14583811	C₂×C₄	semidirect
98. Q₁₆×C₁₇₃	C₂×C₄	semidirect
99. SD₁₆×C₁₇₃	C₂×C₄	semidirect
169. C₁₇₃	#2956	semidirect
186. #137	#10621924	semidirect
238. Q₁₆	C₂×#10621924	semidirect
242. SD₁₆	C₂×#10621924	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26012571	2
#26012570	2
#26012537	2
#26012536	2
#26012535	2
#26012534	2
#26012506	2
C₂×#19149744	2
D₁₇₃⋊#104	2
M₁₆×#10621924	2
#26012290	2
#2956	173